On the Hojman conservation quantities in Cosmology

Abstract

We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like f (R) gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. φ (t) = φ (a (t)), which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for f (T ) teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of f (T ) functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x The source for the late-time cosmic acceleration [1][2][3][4] has been attributed to an unidentified type of matterenergy with a negative parameter in the equation of state, the dark energy. The cosmological constant, Λ, is the simplest candidate for dark energy with a parameter in the equation of state, w Λ = −1. However, the cosmological constant suffers from two major problems. They are the fine tuning and the coincidence problems [5,6]. Consequently other dark energy candidates have been introduced, such as cosmologies with time-varying Λ (t), quintessence, Chaplygin gas, matter creation, f (R) gravity and many others. This status of art imposes a discrimination among the various cosmological models and the search for new approaches to find out exact solutions to be matched with data. In particular, one of the issue is that cosmological models should come out from some first principles in order to be related to some fundamental * Electronic address: anpaliat@phys.uoa.gr † Electronic address: leach.peter@ucy.ac.cy ‡ Electronic address: capozziello@na.infn.it theory.Below we restrict our consideration to a spatially flat Friedmann-Robertson-Walker (FRW) spacetime with line element ds 2 = −dt 2 + a 2 (t) dx 2 + dy 2 + dz 2 .(1)Let us start our considerations from cosmological solutions derived from General Relativity (GR). Standard GR provides us with a set of second-order differential equations. Cons...

Noether symmetries as a geometric criterion to select theories of gravity

“…Consider now a one parameter point transformation and re-express the differential equation (21) in the transformed variables, i.e. H(x,ȳ,ȳ ′ , ...,ȳ (n) ) = 0.…”

“…As a final comment, it is worth noticing that, apart from the Lie point symmetries a , which are the simplest kind of symmetries, there exist several other types of transformations for searching for symmetries in differential equations. In particular, in [19], S. Hojman proposed a conservation theorem where one uses directly the equations of motion, rather than the Lagrangian or the Hamiltonian of a system and, in general, the conserved quantities can be different from those derived from the Noether Symmetry Approach [20,21]. In addition, there exist higher order symmetries, such as contact symmetries; that is, when the equation of motion are invariant under contact transformations, which are defined as one parameter transformations, in the tangent bundle of the associated dynamical system [22].…”

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

“…Apart from the usual way of finding first-integrals from infinitesimal symmetries via the Noether theorem, and a second procedure based on the existence of alternative geometric structures for the description of the vector field providing us a recursion operator, there is a third approach started by Hojman [75] and González-Gascón in [76], which is becoming more and more important because of its applications in f (R)-gravity and FRW cosmology [77][78][79][80][81][82][83]. It was introduced first for divergence-free vector fields in an oriented manifold (M, Ω), in the particular case of a SODE and then generalised to arbitrary SODE vector fields.…”

Conservation laws and exact solutions in Brans–Dicke cosmology with a scalar field

“…There is not unique method to construct conservation laws of dynamical systems. Some of the different approaches which have been applied in cosmological studies are, (a) point symmetries (b) dynamical symmetries [41,42], (c) Cartan symmetries [43], (d) nonlocal symmetries [44][45][46][47] (e) singularity analysis [48], (f) Hojman approach [49,50], (g) Darboux polynomial [51,52], (h) automorphism [53][54][55], (i) Galois group [56].…”